Łukasz Kaiser

Model Checking Games for the Quantitative μ -Calculus

We investigate quantitative extensions of modal logic and the modal μ-calculus, and study the question whether the tight connection between logic and games can be lifted from the qualitative logics to their quantitative counterparts. It turns out that, if the quantitative μ-calculus is defined in an appropriate way respecting the duality properties between the logical operators, then its model checking problem can indeed be characterised by a quantitative variant of parity games. However, these quantitative games have quite different properties than their classical counterparts, in particular they are, in general, not positionally determined. The correspondence between the logic and the games goes both ways: the value of a formula on a quantitative transition system coincides with the value of the associated quantitative game, and conversely, the values of quantitative parity games are definable in the quantitative μ-calculus.

Information Tracking in Games on Graphs

When seeking to coordinate in a game with imperfect information, it is often relevant for a player to know what other players know. Keeping track of the information acquired in a play of infinite duration may, however, lead to infinite hierarchies of higher-order knowledge. We present a construction that makes explicit which higher-order knowledge is relevant in a game and allows us to describe a class of games that admit coordinated winning strategies with finite memory.

Degrees of lookahead in regular infinite games

We study variants of regular infinite games where the strict alternation of moves between the two players is subject to modifications. The second player may postpone a move for a finite number of steps, or, in other words, exploit in his strategy some lookahead on the moves of the opponent. This captures situations in distributed systems, e.g. when buffers are present in communication or when signal transmission between components is deferred. We distinguish strategies with different degrees of lookahead, among them being the continuous and the bounded lookahead strategies. In the first case the lookahead is of finite possibly unbounded size, whereas in the second case it is of bounded size. We show that for regular infinite games the solvability by continuous strategies is decidable, and that a continuous strategy can always be reduced to one of bounded lookahead. Moreover, this lookahead is at most doubly exponential in the size of the parity automaton recognizing the winning condition. We also show that the result fails for non-regular games where the winning condition is given by a context-free ω-language.

New algorithm for weak monadic second-order logic on inductive structures

We present a new algorithm formodel-checkingweakmonadic second-order logic on inductive structures, a class of structures of bounded clique width. Our algorithm directly manipulates formulas and checks them on the structure of interest, thus avoiding both the use of automata and the need to interpret the structure in the binary tree. In addition to the algorithm, we give a new proof of decidability of weak MSO on inductive structures which follows Shelahs composition method. Generalizing this proof technique, we obtain decidability of weak MSO extended with the unbounding quantifier on the binary tree, which was open before.

Experiments with reduction finding

Reductions are perhaps the most useful tool in complexity theory and, naturally, it is in general undecidable to determine whether a reduction exists between two given decision problems. However, asking for a reduction on inputs of bounded size is essentially a

Solving counter parity games

\Sigma^p_2

Synthesis for Structure Rewriting Systems

problem and can in principle be solved by ASP, QBF, or by iterated calls to SAT solvers. We describe our experiences developing and benchmarking automatic reduction finders. We created a dedicated reduction finder that does counter-example guided abstraction refinement by iteratively calling either a SAT solver or BDD package. We benchmark its performance with different SAT solvers and report the tradeoffs between the SAT and BDD approaches. Further, we compare this reduction finder with the direct approach using a number of QBF and ASP solvers. We describe the tradeoffs between the QBF and ASP approaches and show which solvers perform best on our

Directed graphs of entanglement two

\Sigma^p_2

Game quantification on automatic structures and hierarchical model checking games

instances. It turns out that even state-of-the-art solvers leave a large room for improvement on problems of this kind. We thus provide our instances as a benchmark for future work on

Model checking the quantitative µ-calculus on linear hybrid systems

\Sigma^p_2